Investigating the Ultraviolet Properties of Gravity with a Wilsonian RG Equation

This paper may be slow to get the attention it merits because of its density and length. I have just printed out pages 1-5, and pages 64-70, from the introduction and conclusion sections, to chew over at leisure

"We review and extend in several directions recent results on the asymptotic safety approach to quantum gravity. The central issue in this approach is the search of a Fixed Point having suitable properties, and the tool that is used is a type of Wilsonian renormalization group equation. We begin by discussing various cutoff schemes, i.e. ways of implementing the Wilsonian cutoff procedure. We compare the beta functions of the gravitational couplings obtained with different schemes, studying first the contribution of matter fields and then the so-called Einstein-Hilbert truncation, where only the cosmological constant and Newton's constant are retained. In this context we make connection with old results, in particular we reproduce the results of the epsilon expansion and the perturbative one loop divergences. We then apply the Renormalization Group to higher derivative gravity. In the case of a general action quadratic in curvature we recover, within certain approximations, the known asymptotic freedom of the four-derivative terms, while Newton's constant and the cosmological constant have a nontrivial fixed point. In the case of actions that are polynomials in the scalar curvature of degree up to eight we find that the theory has a fixed point with three UV-attractive directions, so that the requirement of having a continuum limit constrains the couplings to lie in a three-dimensional subspace, whose equation is explicitly given. We emphasize throughout the difference between scheme-dependent and scheme-independent results, and provide several examples of the fact that only dimensionless couplings can have 'universal' behavior."

This quarter (April - June 2008) has been unusually productive of quantum gravity research and there are more good papers out there than one can properly take in. I may be forced to lay this one aside for a few days. Perhaps someone else may take an interest in it and discuss it. If not, I'll get around to it later. I think it is a signficant paper because it provides additional evidence that gravity is actually renormalizable (in a certain special sense*), due to the existence of a fixed point of the renormalization group flow.

Sample exerpts
==quote==
VIII. CONCLUSIONS
In this paper we have reviewed and extended recent work on the asymptotic safety approach to quantum gravity. The central hypothesis of this approach is the existence of a nontrivial FP for gravity, having finitely many UV–attractive directions. Accordingly, most of the work has gone towards proving the existence of such a FP. Let us summarize the evidence that has been obtained by applying Wilsonian RG methods to gravity (for general reviews see also [54, 55, 56]). In order to start from the simplest...
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This may at least in part explain the robustness of the results. When many terms are taken into account in the truncation, it is hard to have an intuitive feeling for the mechanism that gives rise to the FP. For example, the beta functions which are obtained by
taking derivatives of (113) with respect to curvature are exceedingly complicated. In fact, they are manipulated by the software and one does not even see them. This is why we have strived in the first few sections of this paper to emphasize the simplest approximations.

They give a clear and intuitive picture suggesting the emergence of a FP to all orders in the derivative expansion. We would therefore like to conclude by overturning a common belief: the existence of a nontrivial FP does not require a delicate cancellation of terms.
The FP appears essentially due to the dimensionful nature of the coupling constants, and it can be seen already in the perturbative Einstein–Hilbert flow (i.e. in the approximation where one considers just the contribution of gravitons or matter fields with kinetic operators of the form −∇2 + E, where E is linear in curvature). More advanced approximations dress up this simple result with RG improvements and with the contribution of additional couplings. The argument in the preceding paragraph suggests that the new couplings will not qualitatively change the results. And indeed, so far it seems that generically such dressing does not spoil the FP. So, to conclude on an optimistic note, one could say that it would actually require a special conspiracy by the new terms to undo the perturbative FP.
==endquote==

More about this is explained in a pedagogical survey paper from last year by Roberto Percacci, which is being published as a chapter in the forthcoming book Approaches to Quantum Gravity. I will get the preprint. The title is Asymptotic Safety.

http://arxiv.org/abs/0709.3851Asymptotic Safety
R. Percacci
To appear in "Approaches to Quantum Gravity: Towards a New Understanding of Space, Time and Matter", ed. D. Oriti, Cambridge University Press
(Submitted on 24 Sep 2007)

"Asymptotic safety is a set of conditions, based on the existence of a nontrivial fixed point for the renormalization group flow, which would make a quantum field theory consistent up to arbitrarily high energies. After introducing the basic ideas of this approach, I review the present evidence in favor of an asymptotically safe quantum field theory of gravity."

What I understand Percacci to be saying here is that thanks to the (apparent) existence of the fixed point, one can use the theory to make calculations and to predict the results of experiment as soon as the values of a few (like three or four) parameters are determined, and can do so without being troubled by infinities. The theory is UV finite and predictive, then.

In other words, it may not be renormalizable in a conventional sense (hence some may object to the researchers occasionally using that term) but it has the finiteness and predictivity and computability which one associates with being renormalizable.

This, as I understand it, is the special sense in which the Asymptotic Safety researchers sometimes speak of gravity as being in a certain way renormalizable. Good to arbitrarily high energy, or down to arbitrarily small scale. No UV cutoff needed. Once the fixed point has been determined.

It is a formidable paper with 86 pages jam packed with calculations, so although I wanted to get it out for eventual discussion, it is fine if people just want to ignore it for a while. I'll get to it when I have more time.